Abstract: We are interested in the simulation and optimisation of gas transport in networks. The gas flow through pipes can be modelled on the basis of the (isothermal) Euler equations. Further network components are described by purely algebraic equations. Depending on the data and the resulting network dynamics, models of different fidelity can be used in different regions of the network. Using adjoint techniques, we derive model and discretisation error estimators. Here, we apply a first-discretise approach. Based on the time-dependent structure of the considered problems, the adjoint systems feature a special structure and therefore allow for an efficient solution. A strategy that controls model and discretisation errors to maintain the accuracy of the solution is presented. We provide (technical) details of our implementation and give numerical results.